QML estimation with non-summable weight matrices
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QML estimation with non‑summable weight matrices Jakub Olejnik1 · Alicja Olejnik2 Received: 15 October 2018 / Accepted: 8 June 2020 © The Author(s) 2020
Abstract This paper revisits the theory of asymptotic behaviour of the well-known Gaussian Quasi-Maximum Likelihood estimator of parameters in mixed regressive, high-order autoregressive spatial models. We generalise the approach previously published in the econometric literature by weakening the assumptions imposed on the spatial weight matrix. This allows consideration of interaction patterns with a potentially larger degree of spatial dependence. Moreover, we broaden the class of admissible distributions of model residuals. As an example application of our new asymptotic analysis we also consider the large sample behaviour of a general group effects design. Keywords Spatial autoregression · Quasi-maximum likelihood estimation · Highorder SAR model · Asymptotic analysis · Non-summable matrices JEL Classification C21 · C23 · C51
1 Introduction It is a broadly employed assumption in a wide range of theoretical studies on spatial econometrics that the spatial weight matrix is absolutely row and column summable. This restriction is mostly a result of the Central Limit Theorem (CLT) used in the derivation of the result on asymptotic behaviour. Historically, it can be traced to the works of Kelejian and Prucha, e.g. Kelejian and Prucha (2001), who were Electronic supplementary material The online version of this article (https://doi.org/10.1007/s1010 9-020-00326-2) contains supplementary material, which is available to authorized users. * Jakub Olejnik [email protected] Alicja Olejnik [email protected] 1
Faculty of Mathematics and Computer Science, University of Lodz, Lodz, Poland
2
Faculty of Economics and Sociology, University of Lodz, Lodz, Poland
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J. Olejnik, A. Olejnik
first to formulate their assumptions as explicit requirements regarding the spatial weight matrix. Their CLT, which turned out to be a milestone in the development of asymptotic theories for spatial econometric models, relies on the absolute summability of the weight matrix involved. In our study we attempt to reconsider this approach, and we focus specifically on the Quasi Maximum Likelihood (QML) estimator for the spatial autoregressive model. By revisiting the classical argument of Lee (2004) and, importantly, introducing a generalised CLT for linear-quadratic forms, we are able to provide a theory for consistency and asymptotic normality of QML estimates for high-order spatial autoregressive models under relaxed conditions. In particular, our approach allows for spatial weight matrices that, even if row-standardised, may not be absolutely column summable. The standard approach, with the absolute summability requirement on the weight matrix, undoubtedly has the appeal of a simple, self-contained theory. Although there might be a perception that the constraint is necessary for showing the desired asymptotic behaviour of various esti
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